Answer
$(-1)^n \sin \theta $
Work Step by Step
By using the Sum and Difference formulas: $\cos (a+b)=\cos a \cos b -\sin a \sin b$ and $\cos (a-b)=\cos a \cos b +\sin a \sin b$
Recall that the sine of a multiple of $\pi$ is always $0$, and the cosine of a multiple of $\pi$ is always $1$ when $n$ is even and $-1$ when $n$ is odd.
Therefore, $\sin n \pi \cos \theta +\cos n \pi \sin \theta$
or, $ =(0) \cos \theta +\cos n \pi \sin \theta$
or, $ =\cos n \pi \sin \theta$
or, $=(-1)^n \sin \theta $
